3d rossby wave frequency dispersion relation

Consider the linearized, quasi-geostrophic vorticity equation on the beta-plane,
with a mean background flow umean , which allows for vertical structure and propagation of Rossby waves:
(d/dt+umean*d/dx)=(nabla2 ψ+(f02/N2)*(d2 ψ/dz2)+bv

where ψ is the horizontal streamfunction, f0 is the Coriolis parameter at the reference latitude, N2 the Brunt-Vaisala frequency, b is the beta parameter, and v is the meridional velocity component given by v =dψ/dx.
Assume a solution of the form ψ(x,y,z,t) = ψ 0 e i(kx+my+nz-_t) where k, m, n are the wavenumbers in the x, y, z directions, respectively.

Show that the frequency dispersion relation is given by:
ω=umean-b/(k2+m2+n2(f02/N2))

2. Relevant equations

3. The attempt at a solution
tried many times but i actualy have no true direction